What's the big deal with interacting tachyons? I understand that any spacelike path appears to move backward in time from some reference frames, so e.g. a "reflected" tachyon can be absorbed before it was emitted.  But I don't see how this really poses an issue for physics, even if you could consciously send signals back in time.  In the big picture, the physical universe consists of spacetime and some fields and/or particles, some patches of which happen to correspond to our physical bodies, and their combined global configuration over all spacetime appears to satisfy some general patterns we call the laws of physics.  But as long as we believe that global configuration does exist in some immutable form, then why can't it contain, in a localized region, a signal being sent back in time?  We, the signal-senders from the future, don't have to worry about the consequences of such a signal, because all those consequences have already happened, and they brought us to the point of sending a signal back in time.
I feel like that's all fairly obvious reasoning, but then how come Wikipedia for example is harping on about the grandfather paradox and so on?
 A: Our current understanding of almost everything except gravity is based on relativistic quantum field theory (RQFT). This answer explains how tachyons are excluded by a general feature of RQFT.
By tachyon, I mean a particle that travels faster than light. In the case of a classical point particle, that means the particle's worldline is spacelike. But our current understanding of nature is based on quantum physics, so we need to use a definition that makes sense in quantum physics.
Sometimes the name "tachyon" is used for a spacelike line in a Feynman diagram or for a quadratic term in a lagrangian with a negative coefficient, but those things are only superficial. The question is whether or not RQFT allows particles that can be observed traveling faster than the vacuum speed of light. For this, we should focus on the theory's observables.
The core principles of quantum theory are expressed in terms of operators on a Hilbert space, and an observable is an operator that represents something measurable. In QFT, observables are tied to regions of spacetime, not to particles. That's okay, because that's how real experiments with subatomic particles work anyway: we use devices that count the number (zero, one, etc) of particles in a given region of space at a given time. Different kinds of devices are sentitive to different kinds of particles. To address the question about tachyons, we can consider relationships between observables localized in spacelike-separated regions.
Let $A$ and $B$ be two regions of spacetime, separated from each other by a nonzero spacelike interval. How can we diagnose the propagation of a particle from $A$ to $B$ (or from $B$ to $A$, since the sequence of spacelike-separated events is frame-dependent)? A necessary condition is that some observables in $A$ and $B$ are correlated with each other, but that's not a sufficient condition, because correlation does not imply causation. Correlations between $A$ and $B$ can be produced using slower-than-light propagation from any region $C$ that is in both of their past light-cones.
In fact, in RQFT, every state of finite energy has entanglement (and therefore correlations) between some observables localized in $A$ and $B$. This is reviewed in arXiv:1803.04993. This is related to the fact that the energy operator (Hamiltonian) is not localized in any finite region of space. We can't use mere correlations to diagnose the existence of tachyons.
To address the question about tachyons, we can appeal to the split property, a characteristic feature of RQFT. The split property says that observables localized in $A$ and $B$ are completely independent of each other. More precisely, it says that the Hilbert space might as well be a tensor product ${\cal H}_A\otimes {\cal H}_B$ where observables in $A$ and $B$ act only on ${\cal H}_A$ and ${\cal H}_B$, respectively. The split property is reviewed on pages 22-23 in arXiv:1810.05338, which says "We are not aware of any [relativistic] quantum field theory that does not obey the split property [in Minkowski spacetime]." It has been proven in some cases, and it's relatively obvious in constructions based on discrete spacetime (like lattice QED and lattice QCD), so for this answer I'll treat it as a characteristic property of RQFT.
Tachyons are not compatible with the split property. The reason is simple. Mathematically, in a Hilbert space of the form ${\cal H}_A\otimes {\cal H}_B$, correlations between the two factors cannot be produced by operators that act on ${\cal H}_A$ and ${\cal H}_B$ alone. To produce such correlations, we would need to use operators that act on both factors together.
RQFT does have operators that act on both factors together. They're not localized in $A$ or $B$, but any region $C$ inside the past light-cones of both $A$ and $B$ does include such operators. In fact, another general feature of RQFT called local primitive causality says that observables in the given region are completely dependent on observables in the region's past light-cone. More precisely, observables in a given region can be expressed — using algebra and limits — in terms of observables in the past light-cone of that region. (This is usually enforced by taking the theory's equations of motion to be hyperbolic differential equations, like Maxwell's equations and the Dirac equation.) Contrast this with the split property, which says that observables in spacelike-separated regions are completely independent of each other. Local primitive causality says that cause-and-effect relationships can exist between timelike- or lightlike-separated regions, and the split property says that such relationships can't exist between spacelike-separated regions.
Is the split property necessarily a fundamental property of any physically relatistic theory? No, not even in mainstream physics. The split property is not compatible with the holographic principle, which seems to be required in any good quantum theory of gravity. But the holographic principle also indicates that the usual concept of locally-Minkowski spacetime is only an approximation, so questions about tachyons probably don't have any meaning in such a context. If we stick with ordinary RQFT where the question is meaningful and the split property holds, then tachyons are disallowed.
